1. Find the value for c for which y=x+c is a tangent to y=3-x-5x^2
don't understand what to do
You gotta find first the intersection of the line and the curve, so
$\displaystyle x+c=3-x-5x^2\implies5x^2+2x+(c-3)=0.$
Now, so that the line be tangent to the curve, the discriminant of the previous equation must be zero, so $\displaystyle 4-4\cdot5(c-3)=0\implies1-5(c-3)=0.$
Solvin' for $\displaystyle c$ yields the desired answer.
The system formed by the two equations must have an unique solution.
Plug y from the first equation in the second and you'll obtain a quadratic equation. Put the condition that $\displaystyle \triangle =0$, where $\displaystyle \triangle =b^2-4ac$ (a,b,c are the coefficients of the quadratic equation).